Prove that a point belongs to the lock of if and only if is a interior point or a frontier point of .

Some translations are in order here.
‘lock’ must mean closure; ‘frontier’ must mean boundary.

A point is in the closure of the set iff every open set containing the point contains a point of the set.

Thus, if we have a interior point or a frontier point of then by definition it is in the closure.

Suppose , the closure.
If is an interior point of we are done.
So what does it mean to say that is not an interior point?
Now you need to consider two cases: .
Both cases should force to be a boundary point.